2.2 Limits Involving Infinity Hoh Rainforest, Olympic National Park, WA.
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2.2 Limits Involving Infinity
Hoh Rainforest, Olympic National Park, WA
Limits & Infinity
• Considerx
xf1
)(
What happens as x gets to be a big number?
Bigxf
1)( Smallxf )(
What happens as x gets even bigger?
BigVeryxf
1)( SmallVeryxf )(
Limits & Infinity
• Considerx
xf1
)(
What if x is wicked huge?
hugewickedxf
1)( SmallWickedxf )(
What if x equals infinity?
1)(xf 0)( xf
1f x
x
1lim 0x x
As the denominator gets larger, the value of the fraction gets smaller.
There is a horizontal asymptote if:
limx
f x b
or limx
f x b
Horizontal Asymptotes
2lim
1x
x
x 2limx
x
x
This number becomes insignificant as .x
limx
x
x 1
There is a horizontal asymptote at 1.
Horizontal Asymptotes
12lim
x
xx
Horizontal Asymptotes
25
3lim
2
2
x
xx
74
7lim
2
x
xx
sin xf x
x
sinlimx
x
x Find:
When we graph this function, the limit appears to be zero.1 sin 1x
so for :0x 1 sin 1x
x x x
1 sin 1lim lim limx x x
x
x x x
sin0 lim 0
x
x
x
by the sandwich theorem:
sinlim 0x
x
x
Horizontal Asymptotes
5 sinlimx
x x
x
Find:
5 sinlimx
x x
x x
sinlim 5 limx x
x
x
5 0
5
Horizontal Asymptotes
Limits & Infinity
• Considerx
xf1
)(
What if x is wicked small?
smallwickedxf
1)( HugeWickedxf )(
What if x is an infinitely small positive number?
0
1)( xf )(xf
Whenever a denominator of a fraction approaches zero, the fraction equals +/ - infinity
1f x
x
0
1limx x
As the denominator approaches zero, the value of the fraction gets very large.
If the denominator is positive then the fraction is positive.
0
1limx x
If the denominator is negative then the fraction is negative.
vertical asymptote at x=0.
Vertical Asymptotes:
20
1limx x
20
1limx x
The denominator is positive in both cases, so the limit is the same.
20
1 limx x
Vertical Asymptotes:
Is this really true?DNE
Find the Vertical Asymptotes:
342
1)(
2
xx
xxf
Vertical @ x= 2, 3
To find whether the asymptote goes up or down, you must examine the limits.
)(lim2
xfx
)(lim2
xfx
Find the Vertical Asymptotes:
342
1)(
2
xx
xxf
Vertical @ x= 2, 3
To find whether the asymptote goes up or down, you must examine the limits.
)(lim3
xfx
)(lim3
xfx
Find the Vertical & Horizontal Asymptotes:
43
56)(
2
2
xx
xxxf
Find the Vertical & Horizontal Asymptotes:
6
65)(
2
2
xx
xxxf
End behavior models model the behavior of a function as x approaches infinity or negative infinity.
A function g is:
a right end behavior model for f if and only if
lim 1x
f x
g x
a left end behavior model for f if and only if
lim 1x
f x
g x
End Behavior Models:
End Behavior Models:• Find an end behavior model for
252 xxy
End Behavior Models:• Show that y = 3x4 is an end behavior model for • y = 3x4 - 2x2 – 7x +8.
End Behavior Models:• Show that y = 2 is an end behavior model for
21
1
xy
Test ofmodel
Our modelis correct.
xf x x e End Behavior Models
As , approaches zero.x xe(The x term dominates.)
g x x becomes a right-end behavior model.
limx
x
x e
x
lim1
x
x
e
x
1 0 1
xh x e becomes a left-end behavior model.
limx
xx
x e
e
lim 1xx
x
e 0 1 1
As , increases faster than x decreases,x xe
therefore is dominant.xe
Test ofmodel
Our modelis correct.
xf x x e
g x x becomes a right-end behavior model.
xh x e becomes a left-end behavior model.
On your calculator, graph:
1
2
3
x
x
y x
y e
y x e
10 10x
1 9y
Use:
5 4 2
2
2 1
3 5 7
x x xf x
x x
Right-end behavior models give us:
5
2
2
3
x
x
32
3
x
dominant terms in numerator and denominator
Often you can just “think through” limits.
1lim sinx x
0
0lim sinx
x
0
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